For an idea of what may happen this coming summer, have a look
what we did the last time, in illiMath 2006.
Sample LaTeX files,
and related materials accumulate here.
Webpage introducing REU students Summer 2008.
Repository of possibly useful files for
illiMath projects.

illiMath2008: an REU of
Geometrical Visualization in Virtual Environments

Visualization of mathematical phenomena and their transmutations
is a maturing field of geometical computer graphics. Though opportunities
to learn and practice this discipline are now plentiful, few universities have
as long a tradition and as broad a base of application as Illinois.

Students in this program work with
current researchers in diverse areas of mathematics, science and technology,
who use a common programming platform, Syzygy. This distributed graphics
system supports cluster based immersive virtual environments (IVE)
such as the CAVE, the
Beckman Cube, the Krannert CANVAS at UIUC, the DiVE at Duke, and the PORTAL
at the Technical University of Berlin, and the Traveling CANVAS at the
Dennos Museum in Traverse City, Michigan.

In this program students with strong interest and some experience in
mathematical graphics participate in research programs of their
associated mentor (AM). The program provides the facilities (and any necessary
training in their use) for the student to assist their AM in creating visual
materials illustrating research in a variety of media, including
the production of custom software, webpages, animations,
print quality images, etc.

Common to all projects is Syzygy, Python, and OpenGL. Syzygy is the
distributed graphics system, developed by the Integrated Systems Lab (ISL),
for cluster based virtual environments The program also provides
the student with tutorials in the mathematical
background and lectures by specialists in the subjects
of their projects.

Projects:

Projects of illiMath2008 Research Experience for Undegraduates, Mathematics
Department, University of Illinois at Urbana (UIUC), directed by
Prof. George Francis, are the original work of the student principal
investigator (SPI) and their associated mentor(s) (AM) and corresponding
mentor(s) (CM). Other student investigators (SI) assist.

baREUcuda:

Eight innovative and promising experiments involving the NVIDIA GPUs, which are programmable by
the C-like CUDA language, were conducted.
The CUDA-chip is, in effect, a small but powerful single-instruction multiple-data (SIMD) parallel
supercomputer that fits into chassis of a desktop and the limits of a student's budget.
In the early 90s, the Connection Machine (CM2) was the SIMD mainline supercomputer of the NCSA.
It is celebrated in Richard Powers's novel Galatea _2.2 .
It also was the platform on which Stephen Wolfram et al performed their lattice gas cellular automata experiments on the Navier Stokes Equation.

NVIDIA provides a library of provocative example, including "fluidsGL" and "Mandelbrot". Extending
Jared Schaber's Math 198 project of
splitting fluidsGL into a CUDA based data server, and a remote (socketed) OpenGL visualizing client, t
he baREUcuda team achieved the following variants:
(1) CUDA-MathLink-Mathematica connection for fluidsGL and Mandelbrot,
(2) CUDA-OpenGL for SimpleGL and Mandelbrot
(3) CUDA-MathLink-Mathematica connection for FFTs and other
matrix operations in the CUDA library.

illiMath Testing and Documentation:

Without testing and documentation an illiMath project has only a brief existence and an
ignominious passage into oblivion. So this project maintained the high level of testing and
copious documentation of all projects.

Celestial Mechanics:

Celestial Mechanics began as Newton's masterpiece application
of his Infinitesimal Calculus. It
remains a perpetual challenge to present its wonders
to a lay audience. The classical
orrery , a mechanical model of the solar system, has now been replaced
by computer graphics. But technology changes, and for several years now
a splendid orrery by Stuart Levy in the CAVE no longer worked.
What began as a Math 198 project, culminating in the
illiSol and
Voyager
projects, was expanded, and expertly documented in this REU project.

The final component of this project, PartiSZG , implemented Levy's
particle viewer,
PartiView
originally developed for the Hayden Planetarium's Rose Center. PartiSZG
as a Syzygy port of PartiView suitable for showing in virtual environments.

Quasicrystals:

DeBruijn proposed two methods for realizing quasicrystals as projections of
sub-lattices of the 6D unit lattice into 3-space. In this collaboration with
mathematical artists, Tony Robbin, deBruijn dual method is used to create
real-time interactive CUBE animations (RTICA). In the current iteration of this
multi-year project
the code was reformulated into the Master-Slave framework of Syzygy, and extended
to display 3 different spacings of DeBruijn's multigrids, 3 different skins of the
component bricks, and 3 different selections of sub-lattices.

caveMathematica

The signal achievement of this project was to revive caveMathematica.exe
so that this Syzygy front-end for Mathematica again worked reliably on all
Syzygy platforms, including the virtual environments at Beckman and on local
Syzygy phleets (clusters) in the REU Lab. A number of applications were
created by members of this group. CaveMathematica started as a
illiMath2006 REU project by (later) WRI intern Mimi Tsuruga (now Berlin Mathematical School),
mentored by Ulises Cervantes-Pimentel of WRI, and Stuart Levy of the NCSA.

ALICE: A Subriemannian Geometry on Nil:

One of Thurston's eight 3D geometries, the Nilpotent Lie Group, can also be given a
sub-Riemnannian geometry. This is easier to imagine and visualize. The Mathematica
Notebook "subNil.nb" contributes to the grand project
ALICE on the Eight-fold Way
to visualize Thurston's eight possible 3D geometries.

illiNav: Lookahead:

This project was motivated by the common experience of virtual world designers
that novices tend to over-correct the VR navigational control systems. This is
likely due to beginners internalizing the controls as an immediate
stimulus-response system, as in a video-game. Providing the navigator with a
"look-ahead" prediction of what the immediately past strategy will result in
should reduce this steering problem. This old problem from control theory
becomes amenable to the new computational power of VR-simulation.

The success of this first try at such a design was demonstrated
when members of both REU programs flew the CAVE through Ulrich Brehm's
knotbox in a trefoil-knot flight-path without bumping into the surface.

Islamic Mathematics

Though the author of this project was not a paid member of the REU program,
she participated in our seminars, and presented her work on the cultural
underpinnings of medieval Islamic contributions to modern algebra.

GeoProj: Geometric projections from the sphere to the plane:

In the pedagogy of an undergraduate course in non-Euclidean geometry
it is desirable for students to make constructions in all, pairwise
linked models of hyperbolic, spherical and Euclidean geometry.
Currently available plane geometry drawing packages, open source and proprietary,
rarely provide more than two equivalent models, and almost none link the
constructions between pairs of models. GeoProj seeks to remedy this,
using a variety of platforms, including Java, Mathematica, C/C++, and Syzygy.

igl2ogl:

AM: George Francis; SIs: John Pacey, Abdul Dakkak, Lisa Hickok.

Syzygy, like nearly all contemporary graphics systems, is based on the
OpenGL graphics libraries. Historically, Kurt Akeley's OpenGL
is the successor to Jim Clark's graphics libarary, GL,
on which the phonemenal success of his Silicon Graphics company (SGI)
was based. An enormous number of GL applications
were developed in the first 6 years of the CAVEs. Not all of these were
translated from Iris GL (igl) to OpenGL (ogl) in the mid-nineties. Some of
the illiMath legacy programs need to be translated before the last SGI machines
in the REU Lab expire. This project was a modest beginning. It is based on
a semi-automatic process augmented by a minimum of hand intervention.
It was demonstrated to work in the CAVE.